We may define the category of relations in the topos $\ECat$ (written $\REL{\ECat}$) in the following manner:

• we set $\Ob{\REL{\ECat}} = \Ob{\ECat}$;
• an arrow from $X$ to $Y$ in $\REL{\ECat}$ is a subobject of $X\times Y$ in $\ECat$ — more precisely an arrow from $X\times Y$ into $\Omega$;
• the composition of $\phi:X\times Y\to \Omega$ and $\psi:Y\times Z\to\Omega$ is done by taking the fibered product $P$ over $Y$ of the subobjects corresponding to $\phi$ and $\psi$ and taking the characteristic function of the image of $P$ in $X\times Z$.

The verification of the fact that $\REL{\ECat}$ is a category is left to the reader.